Quantitative strong unique continuation for elliptic operators -- application to an inverse spectral problem

TL;DR

This paper develops quantitative measures for the strong unique continuation property of elliptic operators with unbounded coefficients and applies these results to demonstrate gauge equivalence of certain Dirichlet-Laplace-Beltrami operators based on boundary spectral data.

Contribution

It provides a quantitative version of strong unique continuation for elliptic operators with unbounded coefficients and applies it to an inverse spectral problem involving gauge equivalence.

Findings

Quantitative strong unique continuation established for elliptic operators.

Proved gauge equivalence of Dirichlet-Laplace-Beltrami operators under boundary spectral data conditions.

Derived uniform quantitative estimates for eigenfunctions.

Abstract

Based on the three-ball inequality and the doubling inequality established in [23], we quantify the strong unique continuation established by Koch and Tataru [21] for elliptic operators with unbounded lower-order coefficients. We also derive a uniform quantitative strong unique continuation for eigenfunctions that we use to prove that two Dirichlet-Laplace-Beltrami operators are gauge equivalent whenever their corresponding metrics coincide in the vicinity of the boundary and their boundary spectral data coincide on a subset of positive measure.